# Primordial Regular Black Holes: Thermodynamics and Dark Matter

## Abstract

**:**

^{−22}of relativistic matter is required to be converted into primordial black holes (PBHs) in order to explain the present abundance of dark matter particles.

## 1. Introduction

^{6}M

_{⊙}[3]. These observations strongly suggest the presence of a supermassive black hole in the center of the Milky Way, since no other adequate alternative for the nature of such a massive object was proposed up until now. Thus, the existence of “stellar” black holes with masses of few tens of the solar mass, or supermassive black holes with masses of six up to nine orders of magnitude the mass of the Sun, seems to be well established.

^{6}times the Planck mass can be formed at the end of the inflationary epoch, when the oscillations of the inflaton field are intense, and reheating occurs. These newly formed black holes have a short period of growth, and then they evaporate until reaching masses close to the extremal case. Such black hole remnants are possible candidates for dark matter particles. It will be shown that only a small fraction of relativistic matter (3.8 × 10

^{−22}) needs to collapse into black holes in order to explain the present dark matter abundance. The paper is organized as follows. In Section 2, the main properties of the Bardeen and the Hayward black holes are reviewed, while the thermodynamic properties of these objects are discussed in Section 3. Then, in Section 4, the formation of regular black holes in the early universe is considered, and finally, in Section 5, the main results are discussed.

## 2. Regular Black Holes

## 3. Thermodynamics of Regular Black Holes

^{14}g have already disappeared by now. It is worth mentioning that the Hawking radiation reinforces the connection between mechanic and thermodynamic laws, suggesting that the horizon surface should be interpreted as the physical entropy and the surface gravity as the physical temperature of the black hole.

_{B}is defined as:

_{B}or K

_{H}as a function of the horizon radius.

## 4. Primordial Regular Black Holes

**, above which the collapse occurs. Comparing the Jeans and the horizon lengths at the time when the collapsing region breaks away from the Hubble expansion, one finds that the critical density contrast must be of the order of the unity. The second key assumption concerns the final mass of the black hole, which is commonly supposed to be approximately close to the horizon mass at the epoch of formation.**

_{c}_{H}is the horizon mass scale, and K and h are constants. Since the PBH mass goes to zero as the density contrast is close to δ

**, the existence of critical phenomena suggests the possibility that masses at formation could be much smaller than the horizon scale. Recent studies indicate that PBHs with a broad mass spectrum can be formed in the high peaks of the co-moving curvature power spectrum resulting from single field inflation [33]. However, the converted mass fraction into black holes is sensitive to possible non-Gaussianities in the amplitude distribution of such large and rare density fluctuations [34].**

_{c}^{−17}M

_{⊙}or higher than 10 M

_{⊙}. Hence, present astronomical data do not impose any constraint on the existence of Planck mass black holes, and on the interpretation of these objects as dark matter particles. However, an opposite direction has been taken by some authors, who have suggested that dark matter candidates are more massive black holes either with masses of about 30 M

_{⊙}[36] or in the range of 10–10

^{5}M

_{⊙}[37].

#### Lower Limits for PHBs Masses from Thermodynamics

^{12}GeV. Recalling that γ and ${x}_{H}$ are connected by the equation$f(x,\gamma )=0$, the numerical solution of these equations gives $\gamma \approx 1.7\times {10}^{-7}$. The associated black hole mass is:

^{−1}, which corresponds to an accretion timescale of $1.2\times {10}^{-25}\text{\hspace{0.17em}}{\mathrm{s}}^{-1}$. This value is six orders of magnitude smaller than the evaporation timescale derived above, indicating that the newly formed black hole will grow. The scenario is the same for the Bardeen case. However, the accretion rate either for the Hayward or Bardeen black holes depends on the energy density of the cosmic relativistic matter, which varies with the temperature as $\epsilon \propto {T}^{4}$. Due to the fast expansion of the universe, the temperature decreases with a timescale ${t}_{col}=T/\left|dT/dt\right|=1/H$. At reheating, this is about $4.7\times {10}^{-31}$ s, which is several orders of magnitude smaller than the two other timescales. This means that these PBHs have initially a very short phase of growth, and then, the evaporation process dominates until the extreme situation is reached.

^{19}GeV/c

^{2}), implying presently that only a low-particle density is required to explain the observations.

## 5. Discussions

^{2}[46], or on the contrary, having masses around few TeV/c

^{2}, resulting from the SO(10) breaking [47]. In the present work, the possibility that primordial regular black holes could be identified with dark matter particles was investigated. This possibility is not new, and past studies always had difficulties with the existence or not of remnants left by the evaporation process. Investigations of the gravitational collapse based on LQG suggest the appearance of a non-singular space–time with a Reissner–Nordström-like metric or, in other words, including two horizons. This behavior is well reproduced by regular black holes, whose geometry is described either by the Bardeen or the Hayward metric.

^{12}GeV, the minimum masses are respectively $2.6\times {10}^{6}{M}_{P}$ for the Bardeen solution, and $5.0\times {10}^{6}{M}_{P}$ for the Hayward case. Recently, a similar scenario has been investigated [49], in which black hole formation occurs during the oscillatory phase after inflation in conditions of slow reheating. The authors have estimated that the minimum black hole mass at formation is ${M}_{\mathrm{min}}=4\pi {M}_{P}({M}_{P}/{H}_{*})\sim {10}^{6}{M}_{P}$, since the expansion rate during inflation derived from Planck 2015 is ${H}_{*}\approx {10}^{14}$ GeV. Notice that this value compares quite well with our own estimates based on thermodynamic arguments.

^{19}GeV/c

^{2}, and only a very small fraction of the relativistic matter at reheating is needed to be converted into PBHs in order to explain the observed dark matter abundance, in agreement with the estimates made by Carr et al. [49].

## Conflicts of Interest

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**Figure 1.**Normalized horizon temperature for Bardeen (black curve) and Schwarzschild (red curve) black holes as a function of the horizon radius in units of the Planck scale distance.

**Figure 2.**Normalized horizon temperature for Hayward (black curve) and Schwarzschild (red curve) black holes as a function of the horizon radius in units of the Planck scale distance.

**Figure 3.**Normalized luminosities for the Bardeen (

**left**panel) black hole and the Hayward (

**right**panel) black hole as a function of the horizon radius in units of the Planck distance scale.

**Figure 4.**Variation of the critical radius in units of the gravitational radius as a function of the Hayward black hole mass in units of the Planck mass (black curve). The radial component of the four-velocity of the flow is also shown as a function of the black hole mass (blue curve).

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Pacheco, J.A.d.F.
Primordial Regular Black Holes: Thermodynamics and Dark Matter. *Universe* **2018**, *4*, 62.
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Primordial Regular Black Holes: Thermodynamics and Dark Matter. *Universe*. 2018; 4(5):62.
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**Chicago/Turabian Style**

Pacheco, José Antonio de Freitas.
2018. "Primordial Regular Black Holes: Thermodynamics and Dark Matter" *Universe* 4, no. 5: 62.
https://doi.org/10.3390/universe4050062